Ordinarily, when an object bounces off of a surface — whether it's light reflecting from a mirror or a billiard ball bouncing off the side of a billiards table — its path makes the same angle with the surface before and after the bounce. However, a Bizarro Billiards table behaves differently. The table is a rectangle with two horizontal and two vertical sides in the x–y plane. The rule that determines how balls bounce is: If the ball is moving up and right along a line with slope 1, and it hits the top side of the table, it bounces off and continues moving down and right along a line with slope −1/2. If the ball is moving up and right along a line with slope 2, and it hits the top side of the table, it bounces off and continues moving down and right along a line with slope −1. These two bounces are reversible: if the ball is moving up and left along a line with slope −1/2 or −1, it bounces off and continues moving down and left along a line with slope 1 or slope 2, respectively. When the ball is bouncing off another side of the table, the rule for bouncing is the same as it would be if you rotated the table to make that side the top side. Suppose that the ball starts in the top left corner (point B) moving down and right along a line with slope −1. If the ball hits side AD and bounces off, then hits side BC and bounces off, and then ends up at corner D, what must the proportions of the rectangle be? Suppose that the ball starts in the bottom left corner (point A) moving up and right along a line with slope 1. If the ball hits side BC and bounces off, then hits side CD and bounces off, then hits side AD and bounces off, and then ends up at corner B, what must the proportions of the rectangle be? Suppose that the rectangle has height AB = 3 and width BC = 5, and the ball starts in the bottom left corner (point A) moving up and right along a line with slope 1. Describe the trajectory that the ball takes